The reason for this subtlety is that the energy and momentum in the gravitational field cannot be unambiguously localized.
However, the "gravitational field energy" is not a part of the energy–momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity).
While in certain situations it is possible to rewrite the equations so that part of the "gravitational energy" now stands alongside the other source terms in the form of the stress–energy–momentum pseudotensor, this separation is not true for all observers, and there is no general definition for obtaining it.
As it turns out, at least for spacetimes which are asymptotically flat (roughly speaking, which represent some isolated gravitating system in otherwise empty and gravity-free infinite space), the ADM 3+1 split leads to a solution: as in the usual Hamiltonian formalism, the time direction used in that split has an associated energy, which can be integrated up to yield a global quantity known as the ADM mass (or, equivalently, ADM energy).
[3] Alternatively, there is a possibility to define mass for a spacetime that is stationary, in other words, one that has a time-like Killing vector field (which, as a generating field for time, is canonically conjugate to energy); the result is the so-called Komar mass[4][5] Although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes.
[6] The Komar integral definition can also be generalized to non-stationary fields for which there is at least an asymptotic time translation symmetry; imposing a certain gauge condition, one can define the Bondi energy at null infinity.
[5] Great effort has been expended on proving positivity theorems for the masses just defined, not least because positivity, or at least the existence of a lower limit, has a bearing on the more fundamental question of boundedness from below: if there were no lower limit to the energy, then no isolated system would be absolutely stable; there would always be the possibility of a decay to a state of even lower total energy.
[7] While the focus here has been on energy, analogue definitions for global momentum exist; given a field of angular Killing vectors and following the Komar technique, one can also define global angular momentum.
Eventually, the hope is to use a suitable defined quasi-local mass to give a more precise formulation of the hoop conjecture, prove the so-called Penrose inequality for black holes (relating the black hole's mass to the horizon area) and find a quasi-local version of the laws of black hole mechanics.
Because the system has a time translation symmetry, Noether's theorem guarantees that it has a conserved energy.
This is similar to the way that Gauss's law defines the charge enclosed by a surface as the normal electric force multiplied by the area.
Of the two definitions, the description of Komar mass in terms of a time translation symmetry provides the deepest insight.
If a system containing gravitational sources is surrounded by an infinite vacuum region, the geometry of the space-time will tend to approach the flat Minkowski geometry of special relativity at infinity.
For systems in which space-time is asymptotically flat, the ADM and Bondi energy, momentum, and mass can be defined.
In terms of Noether's theorem, the ADM energy, momentum, and mass are defined by the asymptotic symmetries at spatial infinity, and the Bondi energy, momentum, and mass are defined by the asymptotic symmetries at null infinity.
Note that mass is computed as the length of the energy–momentum four-vector, which can be thought of as the energy and momentum of the system "at infinity".
The Einstein summation convention is assumed for repeated indices but the sum over k and j only runs over the spatial directions.
It is also possible to obtain expressions for the momentum in asymptotically flat spacetime.
Translating the above statement into the language of general relativity, we say that a system in nearly flat space-time has a total non-gravitational energy E and momentum P given by: When the components of the momentum vector of the system are zero, i.e. Pi = 0, the approximate mass of the system is just (E+Ebinding)/c2, Ebinding being a negative number representing the Newtonian gravitational self-binding energy.
But using the equations of motion, one can also write this as where the sum over j runs only over the spatial directions and the second equality uses the fact that
This conjecture was soon proved to be correct by one of Hilbert's close associates, Emmy Noether.
However, when the symmetry is an infinite parameter continuous group, the existence of a conserved energy is not guaranteed.
In a similar manner, Noether's theorem associates conserved momenta with space-translations, when the symmetry group of the translations is finite-dimensional.
Noether's theorem proves that such stationary space-times must have an associated conserved energy.
ADM mass was introduced (Arnowitt et al., 1960) from an initial-value formulation of general relativity.
Pseudotensors are not gauge invariant – because of this, they only give consistent gauge-independent answers for the total energy when additional constraints (such as asymptotic flatness) are met.